Lubrication dynamics of a settling plate

Abstract

If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power law: $h\sim t^{-1/2}$ . We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex. We find that, depending upon the initial position of the plate and the position of the centre of force, the plate might either make contact in finite time or settle progressively without ever making contact. Our results show an excellent agreement between analytical exact solutions, asymptotic solutions and numerical studies of the lubrication equations.